Electric field of a horizontal antenna above a homogeneous half‐space: Implications for GPR

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Abstract
Inverse modeling and interpretation of subsurface structures depend on accurate knowledge of the undisturbed field. This is especially true in the analysis of radargrams, in which it is difficult to resolve the upper homogeneous medium from the less shallow scatterers. The available forward models based on plane‐wave and ray approximation are not accurate enough for this task. To improve resolution capabilities, we determine the undisturbed field using exact expressions for the electric field of a sine‐shaped ground‐penetrating radar (GPR) signal antenna above a homogeneous half‐space. In the frequency domain it consists of the sum of two improper integrals with complex integrands. Each integrand contains a kernel multiplied by a Bessel function of the first kind and of order zero or one. In the general case these integrals do not have a solution in closed form, and their integrands are poorly convergent. Therefore, to solve the integrals we must use a special formalism involving integrals around branch points. When we assume that both the transmitter and the receiver are on the boundary of the half‐space, there exist analytic solutions for the first integral without further restrictions and for the second integral for two special cases: free space and half‐space, neglecting displacement currents. We check our corresponding numerical results against these analytic solutions. In the time domain we represent the electric field as a function of transmitter‐receiver offset and time. For a purely dielectric half‐space the backtransformation of the first integral is analytical under the assumed simplification, allowing us to check the numerical results obtained with a fast Fourier transform (FFT) algorithm. These results allowed us to design radargrams for five different models of a homogeneous earth, and they are fundamental for interpretation and further research of GPR modeling.